A056885 -id:A056885 - cp's OEIS frontend

A001752 Expansion of 1/((1+x)*(1-x)^5).

1, 4, 11, 24, 46, 80, 130, 200, 295, 420, 581, 784, 1036, 1344, 1716, 2160, 2685, 3300, 4015, 4840, 5786, 6864, 8086, 9464, 11011, 12740, 14665, 16800, 19160, 21760, 24616, 27744, 31161, 34884, 38931, 43320, 48070, 53200, 58730, 64680, 71071, 77924, 85261

Offset: 0

N. J. A. Sloane

Define a unit column of a binary matrix to be a column with only one 1. a(n) = number of 3 X n binary matrices with 1 unit column up to row and column permutations (if offset is 1). - Vladeta Jovovic, Sep 09 2000
Generally, number of 3 X n binary matrices with k=0,1,2,... unit columns, up to row and column permutations, is the coefficient of x^k in 1/6*(Z(S_n; 5 + 3*x,5 + 3*x^2, ...) + 3*Z(S_n; 3 + x,5 + 3*x^2,3 + x^3,5 + 3*x^4, ...) + 2*Z(S_n; 2,2,5 + 3*x^3,2,2,5 + 3*x^6, ...)), where Z(S_n; x_1,x_2,...,x_n) is the cycle index of symmetric group S_n of degree n.
First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).
Transform of tetrahedral numbers, binomial(n+3,3), under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005
Equals triangle A152205 as an infinite lower triangular matrix * [1, 2, 3, ...]. - Gary W. Adamson, Feb 14 2010
With a leading zero, number of all possible octahedra of any size, formed when intersecting a regular tetrahedron by planes parallel to its sides and dividing its edges into n equal parts. - V.J. Pohjola, Sep 13 2012
With 2 leading zeros and offset 1, the sequence becomes 0,0,1,4,11,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q) with p <= q. Then b(n) is the total volume of the family of rectangular prisms with dimensions p, |q - p| and |q - p|. - Wesley Ivan Hurt, Apr 14 2018
Conjecture: For n > 2, a(n-3) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of an n X n square matrix M(n) defined as the n-th principal submatrix of the array A010751 whose general element is given by M[i,j] = floor((j - i + 1)/2). - Stefano Spezia, Jan 12 2022
Consider the following drawing of the complete graph on n vertices K_n: Vertices 1, 2, ..., n are on a straight line. Any pair of nonconsecutive vertices (i, j) with i < j is connected by a semicircle that goes above the line if i is odd, and below if i is even. With four leading zeros and offset 1, a(n) gives the number of edge crossings of the aforementioned drawing of K_n. - Carlo Francisco E. Adajar, Mar 17 2022

			There are 4 binary 3 X 2 matrices with 1 unit column up to row and column permutations:
  [0 0] [0 0] [0 1] [0 1]
  [0 0] [0 1] [0 1] [0 1]
  [0 1] [1 1] [1 0] [1 1].
For n=5, the numbers of the octahedra, starting from the smallest size, are Te(5)=35, Te(3)=10, and Te(1)=1, the sum being 46. Te denotes the tetrahedral number A000292. - _V.J. Pohjola_, Sep 13 2012

T. A. Saaty, The Minimum Number of Intersections in Complete Graphs, Proc. Natl. Acad. Sci. USA., 52 (1964), 688-690.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 4.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A000217, A000292, A000332, A000929, A002620, A004526, A010751, A056885, A057524, A108561, A152205, A216172, A216173, A216175.

Cf. A001753 (partial sums), A002623 (first differences), A158454 (signed column k=2), A169792 (binomial transform).

[Floor(((n+3)^2-1)*((n+3)^2-3)/48): n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

A001752:=n->(3*(-1)^n+93+168*n+100*n^2+24*n^3+2*n^4)/96:
seq(A001752(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2015

a = {1, 4}; Do[AppendTo[a, a[[n - 2]] + (n*(n + 1)*(n + 2))/6], {n, 3, 10}]; a
(* Number of octahedra *) nnn = 100; Teo[n_] := (n - 1) n (n + 1)/6
Table[Sum[Teo[n - nn], {nn, 0, n - 1, 2}], {n, 1, nnn}]
(* V.J. Pohjola, Sep 13 2012 *)
LinearRecurrence[{4,-5,0,5,-4,1},{1,4,11,24,46,80},50] (* Harvey P. Dale, Feb 07 2019 *)

a(n)=if(n<0,0,((n+3)^2-1)*((n+3)^2-3)/48-if(n%2,1/16))

a(n)=(n^4+12*n^3+50*n^2+84*n+46)\/48 \\ Charles R Greathouse IV, Sep 13 2012

a(n) = floor(((n+3)^2 - 1)*((n+3)^2 - 3)/48).
G.f.: 1/((1+x)*(1-x)^5).
a(n) - 2*a(n-1) + a(n-2) = A002620(n+2).
a(n) + a(n-1) = A000332(n+4).
a(n) - a(n-2) = A000292(n+1).
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+4, 4). - Paul Barry, Jul 01 2003
a(n) = (3*(-1)^n + 93 + 168*n + 100*n^2 + 24*n^3 + 2*n^4)/96. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n-2k+3, 3).
a(n) = Sum_{k=0..n} binomial(k+3, 3)*(1-(-1)^(n+k-1))/2. (End)
a(n) = A108561(n+5,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
From Wesley Ivan Hurt, Apr 01 2015: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 5*(n-4) - 4*a(n-5) + a(n-6).
a(n) = Sum_{i=0..n+3} (n+3-i) * floor(i^2/2)/2. (End)
Boas-Buck recurrence: a(n) = (1/n)*Sum_{p=0..n-1} (5 + (-1)^(n-p))*a(p), n >= 1, a(0) = 1. See the Boas-Buck comment in A158454 (here for the unsigned column k = 2 with offset 0). - Wolfdieter Lang, Aug 10 2017
Convolution of A000217 and A004526. - R. J. Mathar, Mar 29 2018
E.g.f.: ((48 + 147*x + 93*x^2 + 18*x^3 + x^4)*cosh(x) + (45 + 147*x + 93*x^2 + 18*x^3 + x^4)*sinh(x))/48. - Stefano Spezia, Jan 12 2022

Formulae corrected by Bruno Berselli, Sep 13 2012

A057669 Triangle T(n,k) of number of minimal 3-covers of an unlabeled n+3-set that cover k points of that set uniquely (k=3,..,n+3).

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 7, 7, 6, 3, 11, 13, 14, 9, 4, 16, 22, 26, 21, 13, 5, 23, 34, 44, 40, 31, 17, 7, 31, 50, 68, 68, 59, 41, 23, 8, 41, 70, 100, 106, 101, 79, 55, 28, 10, 53, 95, 140, 157, 158, 136, 106, 68, 35, 12, 67, 125, 190, 221, 234, 214, 182, 132, 85, 42, 14, 83, 161

Offset: 0

Vladeta Jovovic, Oct 16 2000

Row sums give A005783.

			[1], [2, 1], [4, 3, 2], [7, 7, 6, 3], ...
There are 7 minimal 3-covers of an unlabeled 6-set that cover 3 points of that set uniquely: {{1}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5}, {3, 4, 5, 6}}, {{1, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5}}, {{1, 5, 6}, {2, 4, 6}, {3, 4, 5, 6}}, {{1, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}, {{1, 4, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}}.

V. Jovovic, More information

Cf. A001752, A056885, A057222, A057223, A057524.

T(n, k) = b(n, k)-b(n-1, k); b(n, k) = coefficient of x^k in x^3/6*(Z(S_n; 5+3*x, 5+3*x^2, ...)+3*Z(S_n; 3+x, 5+3*x^2, 3+x^3, 5+3*x^4, ...)+2*Z(S_n; 2, 2, 5+3*x^3, 2, 2, 5+3*x^6, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

A057222 Number of 4 X n binary matrices with 1 unit column up to row and column permutations.

Original entry on oeis.org

1, 6, 27, 102, 333, 969, 2572, 6309, 14472, 31333, 64500, 127011, 240475, 439626, 778848, 1341286, 2251350, 3691629, 5925443, 9326451, 14417175, 21918490, 32812572, 48422262, 70510271, 101402091, 144137322, 202654565, 282015876, 388677651, 530815688, 718713015, 965220510

Offset: 1

Vladeta Jovovic, Sep 18 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 4-covers of an unlabeled n-set that cover 5 points of that set uniquely (if offset is 5).
Generally, the number b(n, k) of 4 X n binary matrices with k=0, 1, ..., n unit columns, up to row and column permutations, is coefficient of x^k in 1/24*(Z(S_n; 12 + 4*x, 12 + 4*x^2, ... ) + 8*Z(S_n; 3 + x, 3 + x^2, 12 + 4*x^3, 3 + x^4, 3 + x^5, 12 + 4*x^6, ...) + 6*Z(S_n; 6 + 2*x, 12 + 4*x^2, 6 + 2*x^3, 12 + 4*x^4, ...) + 3*Z(S_n; 4, 12 + 4*x^2, 4, 12 + 4*x^4, ...) + 6*Z(S_n; 2, 4, 2, 12 + 4*x^4, 2, 4, 2, 12 + 4*x^8, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f. for b(n,k), k=0,1,..,n, is 1/k!* k - th derivative of 1/24*(1/(1 - x)^12/(1 - x*t)^4 + 8/(1 - x)^3/(1 - x^3)^3/(1 - x^3*t^3)/(1 - x*t) + 6/(1 - x)^6/(1 - x^2)^3/(1 - x^2*t^2)/(1 - x*t)^2 + 3/(1 - x)^4/(1 - x^2)^4/(1 - x^2*t^2)^2 + 6/(1 - x)^2/(1 - x^2)/(1 - x^4)^2/(1 - x^4*t^4)) with respect to t, when t=0.

V. Jovovic, Generating functions

Cf. A057524, A001752, A056885, A057223.

G.f.: 1/6*x*(1/(1-x)^12+2/(1-x^3)^3/(1-x)^3+3/(1-x^2)^3/(1-x)^6).

Added more terms, Joerg Arndt, May 21 2013

A057968 Triangle T(n,k) of numbers of minimal 5-covers of an unlabeled n+5-set that cover k points of that set uniquely (k=5,..,n+5).

Original entry on oeis.org

1, 4, 1, 19, 7, 2, 91, 46, 16, 3, 436, 279, 115, 28, 5, 1991, 1563, 740, 221, 49, 7, 8651, 7978, 4309, 1524, 405, 75, 10, 35354, 37290, 22604, 9272, 2875, 659, 115, 13, 135617, 159948, 107584, 50058, 17840, 4866, 1042, 163, 18, 488312, 633211

Offset: 0

Vladeta Jovovic, Oct 17 2000

Row sums give A005785.

			[1], [4, 1], [19, 7, 2], [91, 46, 16, 3], [436, 279, 115, 28, 5], ...; there are 46 minimal 5-covers of an unlabeled 8-set that cover 6 points of that set uniquely.

More information

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965, A057966(labeled case), A057967.

T(n, k)=b(n, k)-b(n-1, k); b(n, k)=coefficient of x^k in (x^5/5!)*(Z(S_n; 27+5*x, 27+5*x^2, ...)+10*Z(S_n; 13+3*x, 27+5*x^2, 13+3*x^3, 27+5*x^4, ...)+15*Z(S_n; 7+x, 27+5*x^2, 7+x^3, 27+5*x^4, ...)+20*Z(S_n; 6+2*x, 6+2*x^2, 27+5*x^3, 6+2*x^4, 6+2*x^5, 27+5*x^6, ...)+20*Z(S_n; 4, 6+2*x^2, 13+3*x^3, 6+2*x^4, 4, 27+5*x^6, 4, 6+2*x^8, 13+3*x^9, 6+2*x^10, 4, 27+5*x^12, ...)+30*Z(S_n; 3+x, 7+x^2, 3+x^3, 27+5*x^4, 3+x^5, 7+x^6, 3+x^7, 27+5*x^8, ...)+24*Z(S_n; 2, 2, 2, 2, 27+5*x^5, 2, 2, 2, 2, 27+5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

A035347 Triangle of a(n,k) = number of minimal covers of an n-set that cover k points of that set uniquely (n >= 1, k >= 1).

Original entry on oeis.org

1, 0, 2, 0, 3, 5, 0, 6, 28, 15, 0, 10, 190, 210, 52, 0, 15, 1340, 3360, 1506, 203, 0, 21, 9065, 60270, 48321, 10871, 877, 0, 28, 57512, 1132880, 1820056, 636300, 80592, 4140, 0, 36, 344316, 21067452, 76834926, 45455676, 8081928, 618939, 21147, 0, 45

Offset: 1

N. J. A. Sloane

			1; 0,2; 0,3,5; 0,6,28,15; ...

T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.

Cf. A056885 for unlabeled case. Row sums give A046165.

a[n_, k_] := Binomial[n, k] * Sum[ StirlingS2[k, j]*(2^j - j - 1)^(n - k), {j, 1, k}]; a[n_, n_] := Sum[ StirlingS2[n, j], {j, 1, n}]; Flatten[ Table[a[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Jun 26 2012, from formula *)

a(n, k) = C(n, k)*Sum_{j=1..k} S(k, j)*(2^j-j-1)^(n-k), where S(k, j) are Stirling numbers of the second kind.

E.g.f.: Sum_{k>=1} (exp(y*x) - 1)^k/k! * exp((2^k-k-1)x). - Geoffrey Critzer, Jun 28 2013

More terms from Vladeta Jovovic, Sep 06 2000

A057223 Number of 4 X n binary matrices without unit columns up to row and column permutations.

Original entry on oeis.org

1, 4, 14, 44, 127, 335, 830, 1931, 4258, 8943, 17984, 34765, 64873, 117220, 205718, 351552, 586348, 956393, 1528350, 2396631, 3693123, 5599550, 8363304, 12317274, 17904795, 25710327, 36497466, 51255153, 71253960, 98113791, 133885404, 181147299, 243121170, 323807952, 428148174

Offset: 0

Vladeta Jovovic, Sep 18 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 4-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).

V. Jovovic, Generating functions

Cf. A057524, A001752, A056885, A057222.

x='x+O('x^66); Vec(1/24*(1/(1-x)^12 + 8/(1-x)^3/(1-x^3)^3 + 6/(1-x)^6/(1-x^2)^3 + 3/(1-x)^4/(1-x^2)^4 + 6/(1-x)^2/(1-x^2)/(1-x^4)^2)) \\ Joerg Arndt, May 21 2013

1/24*(Z(S_n; 12, 12, ...) + 8*Z(S_n; 3, 3, 12, 3, 3, 12, ...) + 6*Z(S_n; 6, 12, 6, 12, ...) + 3*Z(S_n; 4, 12, 4, 12, ...) + 6*Z(S_n; 2, 4, 2, 12, 2, 4, 2, 12, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f. : 1/24*(1/(1 - x)^12 + 8/(1 - x)^3/(1 - x^3)^3 + 6/(1 - x)^6/(1 - x^2)^3 + 3/(1 - x)^4/(1 - x^2)^4 + 6/(1 - x)^2/(1 - x^2)/(1 - x^4)^2).

Added more terms, Joerg Arndt, May 21 2013

A057967 Triangle T(n,k) of numbers of minimal 4-covers of an unlabeled n+4-set that cover k points of that set uniquely (k=4,..,n+4).

Original entry on oeis.org

1, 3, 1, 10, 5, 2, 30, 21, 11, 3, 83, 75, 49, 18, 5, 208, 231, 177, 84, 30, 6, 495, 636, 554, 318, 143, 42, 9, 1101, 1603, 1540, 1023, 543, 210, 62, 11, 2327, 3737, 3907, 2904, 1759, 822, 311, 82, 15, 4685, 8163, 9153, 7470, 5012, 2706, 1219, 423, 111, 18, 9041

Offset: 0

Vladeta Jovovic, Oct 17 2000

Row sums give A005784.

			[1], [3, 1], [10, 5, 2], [30, 21, 11, 3], [83, 75, 49, 18], ...; there are 5 minimal 4-covers of an unlabeled 6-set that cover 5 points of that set uniquely.

More information

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963, A057964, A057965(labeled case), A057966, A057968.

T(n, k) = b(n, k)-b(n-1, k); b(n, k) = coefficient of x^k in x^4/24*(Z(S_n; 12 + 4*x, 12 + 4*x^2, ...) + 8*Z(S_n; 3 + x, 3 + x^2, 12 + 4*x^3, 3 + x^4, 3 + x^5, 12 + 4*x^6, ...) + 6*Z(S_n; 6 + 2*x, 12 + 4*x^2, 6 + 2*x^3, 12 + 4*x^4, ...)

+ 3*Z(S_n; 4, 12 + 4*x^2, 4, 12 + 4*x^4, ...) + 6*Z(S_n; 2, 4, 2, 12 + 4*x^4, 2, 4, 2, 12 + 4*x^8, ...)), where Z(S_n; x_1, x_2, ..., x_n) is the cycle index of the symmetric group S_n of degree n.

A057972 Number of 5 X n binary matrices with 3 unit columns up to row and column permutations.

Original entry on oeis.org

3, 31, 252, 1776, 11048, 61106, 303664, 1368844, 5651241, 21559133, 76613440, 255411923, 803771681, 2400633464, 6837010458, 18644075466, 48855805143, 123415815229, 301386128354, 713271875603, 1639572164669, 3667859207856

Offset: 3

Vladeta Jovovic, Oct 21 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 8 points of that set uniquely (if offset is 8).

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057969 - A057971.

Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) + 20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f. : x^3/120*(35/(1 - x^1)^27 + 130/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 100/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).

A057969 5 x n binary matrices without unit columns up to row and column permutations.

Original entry on oeis.org

1, 5, 24, 115, 551, 2542, 11193, 46547, 182164, 670476, 2325506, 7624434, 23716419, 70253721, 198905506, 540079754, 1410786483, 3555443969, 8667153126, 20484365167, 47037898503, 105143200252, 229178029000

Offset: 0

Vladeta Jovovic, Oct 20 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5-covers of an unlabeled n-set that cover 5 points of that set uniquely (if offset is 5).

Vladeta Jovovic, The number of minimal covers of an unlabeled n-set that cover k points of that set uniquely
Vladeta Jovovic, Number of binary matrices with fixed number of unit columns up to row and column permutations

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963-A057968, A057970-A057972.

a(n)=(1/5!)*(Z(S_n; 27, 27, ...) + 10*Z(S_n; 13, 27, 13, 27, ...) + 15*Z(S_n; 7, 27, 7, 27, ...) + 20*Z(S_n; 6, 6, 27, 6, 6, 27, ...) + 20*Z(S_n; 4, 6, 13, 6, 4, 27, 4, 6, 13, 6, 4, 27, ...) + 30*Z(S_n; 3, 7, 3, 27, 3, 7, 3, 27, ...) + 24*Z(S_n; 2, 2, 2, 2, 27, 2, 2, 2, 2, 27, ...)), where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.

G.f. : 1/120*(1/(1 - x^1)^27 + 10/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 20/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5 + 24/(1 - x^1)^2/(1 - x^5)^5).

A057970 5 x n binary matrices with 1 unit column up to row and column permutations.

Original entry on oeis.org

1, 8, 54, 333, 1896, 9874, 47164, 207112, 840323, 3168506, 11170331, 37034409, 116095018, 345785753, 982835676, 2676217504, 7005306389, 17681946594, 43153532167, 102080966243, 234565062960, 524594120393, 1143910860870

Offset: 1

Vladeta Jovovic, Oct 21 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 6 points of that set uniquely (if offset is 6).

Vladeta Jovovic, Number of minimal covers of an unlabeled n - set that cover k points of that set uniquely
Vladeta Jovovic, Number of binary matrices with fixed number of unit columns up to row and column permutations

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963 - A057968, A057970 - A057972, A057969, A057971, A057972.

G.f.: x/120*(5/(1 - x^1)^27 + 30/(1 - x^1)^13/(1 - x^2)^7 + 15/(1 - x^1)^7/(1 - x^2)^10 + 40/(1 - x^1)^6/(1 - x^3)^7 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).

cp's OEIS Frontend

A001752 Expansion of 1/((1+x)*(1-x)^5).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A057669 Triangle T(n,k) of number of minimal 3-covers of an unlabeled n+3-set that cover k points of that set uniquely (k=3,..,n+3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A057222 Number of 4 X n binary matrices with 1 unit column up to row and column permutations.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A057968 Triangle T(n,k) of numbers of minimal 5-covers of an unlabeled n+5-set that cover k points of that set uniquely (k=5,..,n+5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A035347 Triangle of a(n,k) = number of minimal covers of an n-set that cover k points of that set uniquely (n >= 1, k >= 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A057223 Number of 4 X n binary matrices without unit columns up to row and column permutations.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A057967 Triangle T(n,k) of numbers of minimal 4-covers of an unlabeled n+4-set that cover k points of that set uniquely (k=4,..,n+4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A057972 Number of 5 X n binary matrices with 3 unit columns up to row and column permutations.

Views

Author

Keywords